Question: Alyssa is an ecologist who studies the change in the fox population of the Arctic circle over time. She observed that the population loses $\dfrac{1}{18}$ of its size every $2$ months. The population of foxes can be modeled by a function, $P$, which depends on the amount of time, $t$ (in months). When Alyssa began the study, she observed that there were $185{,}000$ foxes in the Arctic circle. Write a function that models the population of the foxes $t$ months since the beginning of Alyssa's study. $P(t) = $
Explanation: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial number of foxes is $185{,}000$, and that the population loses $\dfrac{1}{18}$ of its size every $2$ months. Note that losing $\dfrac{1}{18}$ is the same as being multiplied by $\dfrac{17}{18}$. [Why?] This means that the initial quantity is $A=185{,}000$ and the factor is $B=\dfrac{17}{18}$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $\dfrac{17}{18}$ every $2$ months. Finding the expression in the exponent We know that the population of foxes is multiplied by $\dfrac{17}{18}$ every $2$ months. This means that each time $t$ increases by $2$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{2}$. When the initial measurement is made, the population hasn't changed. So $P(0) = 185{,}000$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{2}$. Summary We found that the following function models the population of foxes $t$ months since the beginning of Alyssa's study. P ( t ) = 185,000 ⋅ ( 17 18 ) t 2 P(t)=185{,}000\cdot \left(\dfrac{17}{18}\right)\^{ \frac{t}{2}}